Bias Formulas for External Adjustment and Sensitivity Analysis of Unmeasured Confounders

Abstract

Uncontrolled confounders are an important source of bias in epidemiologic studies. The authors review and derive a set of parallel simple formulas for bias factors in the risk difference, risk ratio, and odds ratio from studies with an unmeasured polytomous confounder and a dichotomous exposure and outcome. The authors show how the bias formulas are related to and are sometimes simpler than earlier formulas. The article contains three examples, including a Monte Carlo sensitivity analysis of a preadjusted or conditional estimate. All the bias expressions can be given parallel formulations as the difference or ratio of (i) the sum across confounder strata of each exposure-stratified confounder-outcome effect measure multiplied by the confounder prevalences among the exposed and (ii) the sum across confounder strata of the same effect measure multiplied by the confounder prevalences among the unexposed. The basic formulas can be applied to scenarios with a polytomous confounder, exposure, or outcome. In addition to aiding design and analysis strategies for confounder control, the bias formulas provide a link between classical standardization decompositions of demography and classical bias formulas of epidemiology. They are also useful in constructing general programs for sensitivity analysis and more elaborate probabilistic risk analyses.